
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
July 10th, 2014, 10:40 AM  #1 
Newbie Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0  relationship between derivative of function and derivative of inverse
from dx/dy = 1/y', derive d^2/dy^2 = y''/(y')^3 and d^3x/dy^3 = 3(y'')^2y'y''/(y')^5. I have no idea where to start please explain. Also can you please explain why the second and higher derivatives are d^nx/dy^n and not dx^n/dy^n? 
July 10th, 2014, 04:38 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 Math Focus: Mainly analysis and algebra 
$\newcommand{\d}{\mathbb{d}}$ $$\text{From} \qquad \frac{\d x}{\d y} = \frac{1}{y^\prime} \qquad \text{derive} \qquad \frac{\d^2x}{\d y^2} \qquad \text{and} \qquad \frac{\d^3x}{\d y^3}$$ We just need to differentiate, but note that $y$ is a function of $x$, and in particular $y^\prime$ is a function of $x$. Thus to differentiate with respect to $y$ we need to use the chain rule by writing \begin{align*} \frac{\d x}{\d y} &= \frac{1}{y^\prime(x(y))} &&= {y^\prime}^{1}(x(y)) \\ \frac{\d^2x}{\d y^2} &= {y^\prime}^{2}(x(y)) y^{\prime\prime}(x(y)) \frac{\d x}{\d y} &&= {y^\prime}^{3}(x(y)) y^{\prime\prime}(x(y)) &&= \frac{y^{\prime\prime}}{{y^\prime}^{3}} \\ \frac{\d^3x}{\d y^3} &= 3{ y^{ \prime\prime } (x(y)) }{ y^\prime(x(y)) }^{4}{ y^{ \prime\prime }(x(y)) }\frac{\d x}{\d y} + { y^\prime(x(y)) }^{3}y^{ \prime\prime\prime }(x(y))\frac{\d x}{\d y} &&= 3{y^{ \prime\prime }(x(y))}^2{ y^\prime(x(y)) }^{5} + { y^\prime(x(y)) }^{4}y^{ \prime\prime\prime }(x(y)) &&= \frac{3{y^{ \prime\prime }}^2 + y^\prime y^{ \prime\prime\prime }}{ { y^\prime }^{5} } \end{align*} We write $\frac{\d^2 y}{\d x^2}$ because $\frac{\d}{\d x}$ is an operator. We are not squaring $\frac{\d y}{\d x}$, but doing the operation of differentiation twice on $y$. 
July 10th, 2014, 07:57 PM  #3 
Newbie Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0 
Thank you very much!


Tags 
derivative, function, inverse, relationship 
Search tags for this page 
how to differntiate nverse function my maths,dx/dy=1/y' derive d^3x/dy^3,formula relationship between derivatve of function and derivative of inverse
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Derivative of the inverse function of f(x) = tan2x  king.oslo  Calculus  3  October 18th, 2013 01:44 AM 
Derivative of Inverse Function help  Fridtjof  Calculus  5  November 3rd, 2011 09:03 AM 
Derivative of Inverse Function  Fridtjof  Calculus  21  October 26th, 2011 04:56 PM 
Derivative of Inverse function  you_of_eh  Calculus  3  January 24th, 2011 01:41 PM 
Derivative of the inverse function of f(x) = tan2x  king.oslo  Algebra  0  December 31st, 1969 04:00 PM 